Alex's Adventures in Numberland - Alex Bellos [22]
While base 60 seems like the product of an extraordinarily fertile imagination, sexagesimal has historical pedigree. It is actually the most ancient base system that we know of.
The simplest form of numerical notation is the tally. It has been used in different forms across the world. The Incas kept count by tying knots on ropes, while cave dwellers painted marks on rocks and, since the invention of wooden furniture, bedposts have – figuratively, at least – been marked with notches. The oldest discovered ‘mathematical artefact’ is believed to be a tally stick: a 35,000-year-old baboon fibula found in a Swaziland cave. The ‘Lebombo bone217; has 29 lines scraped on it, which possibly denote a lunar cycle.
As we saw in the previous chapter, humans can instantly tell the difference between one item and two, between two items and three, but beyond four it gets difficult. This is true of notches as well. For any convenient system of tally-keeping, the tallies need to be grouped. In Britain, tally convention is to mark four vertical lines and then make the fifth a diagonal crossing through them – the so-called ‘five-bar gate’. In South America, the preferred style is for the first four lines to mark a square and the fifth is a diagonal in the square. The Japanese, Chinese and Koreans use a more elaborate method, constructing the character, which means ‘correct’ or ‘proper’. (The next time you have sushi, ask the waiter to show you how he is tallying your dishes.)
Tally systems of the world.
Around 8000 BC a practice of using small clay pieces with markings to refer to objects emerged throughout the ancient world. These tokens primarily recorded numbers of things, such as sheep to be bought and sold. Different clay pieces referred to different objects or numbers of objects. From that moment sheep could be counted without actually being there, which made trade and stock-keeping much easier. It was the birth of what we understand now as numbers.
In the fourth millennium bc in Sumer, an area now in present-day Iraq, this token system evolved into a script in which a pointed reed was pressed into soft clay. Numbers were first represented by circles or fingernail shapes. By around 2700 bc the stylus had a flat edge and the imprints looked rather like bird footprints, with different imprints referring to different numbers. The script, called cuneiform, marked the beginning of the long history of Western writing systems. It is wonderfully ironic to think that literature was a by-product of a numerical notation invented by Mesopotamian accountants.
In cuneiform there were symbols only for 1, 10, 60 and 3600, which means the system was a mixture of base 60 and base ten, as the basic set of cuneiform numbers translates into 1, 10, 60 and 60×60. The question why the Sumerians grouped their numbers in sixties has been described as one of the greatest unresolved mysteries in the history of arithmetic. Some have suggested it was the result of the fusion of two previous systems, with bases five and 12, though no conclusive evidence of this has been found.
The Babylonians, who made great advances in maths and astronomy, embraced the Sumerian sexagesimal base, and later the Egyptians, followed by the Greeks, based their time-counting methods on the Babylonian way – which is why, to this day, there are 60 seconds in a minute and 60 minutes in an hour. We are so used to telling the time in base 60 that we never question it, even though it really is quite unexplained. Revolutionary France, however, wanted to iron out what they saw as an inconsistency in the decimal system. When the National Convention introduced the metric system for weights and measures in 1793, it also tried to decimalize time. A decree was signed establishing that every day would be divided into ten hours, each containing 100 minutes, each of which contained 100 seconds. This worked out neatly, making 100,000 seconds in the day – compared